Space as Symmetry

                          Symmetry Under Transformation

The S3 is a hybrid:  combined puzzle and maze and rolling ball device, where intuition is a prerequisite -- logic is necessary but not sufficient.

Sis an architecture of nested dichotomies:  contradictions of reason and perception, frames of reference, of logic, pattern and process, motion and stasis; nested, confusing distinctive-feature information.  

Sis a dynamic router, a system of programmable binary (0/1) switches, a “flying junction” (an over-and-under junction of tunnels which weave past each other) housed within a cubical exoskeleton with multiple axes of rotation and alternative, manipulator-controlled ball trajectories, offering exit in all orientations.

Instead of viewing space as fixed, a passive arena (simple games, puzzles and mazes), the S3 network architecture uses space and geometry as active participants in the problem universe.

                                         The Dialectic  

                    Potential Energy -- Form and Its Negation

Juxtaposition of square and circle across three dimensions.


Contradiction of pattern;


Contradiction of face / quadrants;  


Divergent symmetry of face pairs; 


Symmetry under transformation.

While the S3 exterior has six faces (three sets of opposing faces), each "quartered" face with one or two holes (mismatched hole quadrants block path exit between S3s), the S3 interior has four intertwined chiral tunnels (each tunnel with independent logic gate orientations) with eight entrance / exits.

In other words, to optimize multiple simultaneous paths the manipulator must conceptually drill down” through nested coordinate systems:

CUBICAL exoskeleton (six faces)  >>  

HOLES quadrant (eight holes)  >>  

TUNNELS gate array (four binary switches)  >>  

TUNNEL (binary switch). 

             Binary Processor (0/1)

(Given that a ball must exit at the lowest Slevel and obey gravity, exit is possible in all orientations.)


                                                                             Tangram / S3

                               S3 "Logic of Dichotomies”  

"… the logic of dichotomies ….  A container shall be provided with holes in such a way that they can be opened and closed.  They are to be open at those places that correspond to a 1 and remain closed at those that correspond to a 0.  Through the opened gates small cubes or marbles are to fall into tracks, through the others nothing.  It [the gate array] is to be shifted from column to column as required."

                                                                            Leibniz, G.W., "De Progressione Dyadica -- Pars 1", 1679

A Cybernetic Machine.

A switch, encompassing the Systems Thinking idea and the operation of wholeness, and the relation of whole and parts, structure of systems and their behavior, laws of regulation, and recursive processes of self-observation and self-reference in the service of reasoning.  

A Leibniz  "germinis machinea rationcinatrix."

The Sis the architectural evolution of programming logic, from linear Tangram to S(Quad Problem)


                                              (four checkered paths cross and exit within a single S3)

programmable / reprogrammable components, interconnects -- sequences of dynamic recursive and conditional binary change of state, each step specified in terms of previous steps which the same function has already calculated.

A demonstration of the necessary conceptual fusion of Cybernetics and Systems Thinking in the face of growing understanding of chaos, and adaptation to accelerating rates of change and complexity of system and environment.   


                                                                                  Tessellating Space

"While in the past, science tried to explain observable phenomena by reducing them to an interplay of elementary units investigable independently of each other, conceptions appear in contemporary science that are concerned with what is somewhat vaguely termed 'wholeness', i.e. problems of organization, phenomena not resolvable into local events, dynamic interactions manifest in difference of behaviour of parts when isolated or in a higher configuration, etc.; in short, 'systems' of various order not understandable by investigation of their respective parts in isolation. Conceptions and problems of this nature have appeared in all branches of science, irrespective of whether inanimate things, living organisms, or social phenomena are the object of study.”

                                                      von Bertalanfy, Ludwig, General Systems Theory: Foundations, 

The S3 is about pattern, the rule which governs a system or phenomenon, exactly like numeric, musical, or visual relationships -- and its simultaneous mental and manual rotation. 

                                Original Gedankenexperiment

                                       Tangram Into 3D

Complex chains of webs -- from planar to spatial environment.    



Tangram -- a topology of axioms (conceptually, a square matrix of sixty-four isosceles triangles:  equivalent polyabolos).

Elastic potential motion:  stored energy of position. 


                                                                                                                          Two Monks Paradox

Stasis vs. motion (square vs.  circleness / rotational symmetry), nested and recursive -- planar.  

Contradiction of action and environment.


Juxtaposition of square and circle, across three dimensions (creating “insideness" and “outsideness” and ball agency).





S -- a topology of synergy

A dynamic router (in Field Programmable Gate Array terms, a programmable logic component terms.) 

An organic and reciprocal switching cycle of cognitive development.  

In Buckminster Fuller’s  Tensegrity (tension / compression) terms: "coordination of thought and physical action, the genesis of geometry, system, and structure.” 

A self-generating, self-sustaining, escalating causal loop, a cycle of cognitive development where Manipulator is both change agent and object of change. 

Transformation.  Equilibrium.


Genesis:  Gregory Bateson (Cybernetician / Ethnographer) Double Bind”  paradox:  contradictory cognitive and perceptual signals / messages at different levels, where acknowledgement of that contradiction is forbidden. 

Norbert Wieners "pathological oscillations”, in the face of Russell's paradox:  an oscillating, intermittent restriction of cognition and perception within a dynamic logic hierarchy.      

Patterns of discovery.


                        Ball Path (trajectory) Packing Problem

                 Sommer Polyhedron:  Porous Space-Filler


                                      Hungry Balls  +  Cube (jello) = Tunnels

Configure four tunnels, of unlike symmetry, to allow a ball to enter and exit in any of twenty-four perpendicular orientations, where at least one low-quadrant exit is always possible.  

                                      In Calculus Terms

Four tunnels (trajectories) are as closely packed as possible within a cube (they attempt to “kiss” each other and the faces of the cube) without overlapping and / or kinking and are configured to allow a ball to enter and exit in any of twenty-four perpendicular orientations, with the constraints of gravity and / or specific exits.  In other words:

Problem.   “Given [four sets of] two points [tunnel openings in a hollow cube] A and B in a vertical [inclined] plane, what is the curve [four curves, i.e., four tunnels, closely packed and intertwined in harmony] traced out by a point [ball] acted on only by gravity, which starts at A and reaches B in the shortest [longest] time?”    

                                            Bernoulli's Brachistochrone problem [with a twist], calculus of variations.

                                                                                     Note Paul J.Nahin, When Least is Best (2004)

Solution.    S3

“Description:  The faces of the modular Sommer Cube (S3) feed into twisted conduits that comprise a three-dimensional, tunnelling maze. The website provides example arrangements and construction questions such as "Given that a ball must exit at the lowest S3 level and obey gravity, what is the minimum number of cubes necessary to make four quadruple path cubes, or S3s which utilize all four conduits as passageways?" Available with transparent or opaque walls.”

               “Levels:  High School (9-12), College, Research

        Languages:  English

Resource Types:  Games, Manipulatives

      Math Topics:  Higher-Dimensional Geometry, Topology”

Cybernetics / Systems Thinking


    Tangram Integrated within Archimedes' sphere and cylinder.




                                                                Archimedes’ Tombstone

(The ratio of the volume of a sphere to the volume of the cylinder that contains it is 2:3; the ratio of the surface area of a sphere to the surface area of the cylinder that contains it, plus its circular ends, is also 2:3.) 

© Michael S. Sommer, Ph.D. 2015